U.S. patent number 6,845,671 [Application Number 10/263,294] was granted by the patent office on 2005-01-25 for inverse method to estimate the properties of a flexural beam and the corresponding boundary parameters.
This patent grant is currently assigned to The United States of America as represented by the Secretary of the Navy. Invention is credited to Andrew J Hull.
United States Patent |
6,845,671 |
Hull |
January 25, 2005 |
Inverse method to estimate the properties of a flexural beam and
the corresponding boundary parameters
Abstract
A system and method is used for estimating the properties of a
flexural beam. The beam is shaken transverse to its longitudinal
axis. Seven frequency domain transfer functions of displacement are
measured at spaced apart locations along the beam. The seven
transfer functions are combined to yield closed form values of the
flexural wavenumber in propagation coefficients at any test
frequency.
Inventors: |
Hull; Andrew J (Newport,
RI) |
Assignee: |
The United States of America as
represented by the Secretary of the Navy (Washington,
DC)
|
Family
ID: |
32041971 |
Appl.
No.: |
10/263,294 |
Filed: |
September 27, 2002 |
Current U.S.
Class: |
73/574; 73/581;
73/789 |
Current CPC
Class: |
G01N
3/32 (20130101); G01N 29/075 (20130101); G01N
29/4418 (20130101); G01N 29/52 (20130101); G01N
29/46 (20130101); G01N 2291/02827 (20130101); G01N
2203/0023 (20130101) |
Current International
Class: |
G01N
29/52 (20060101); G01N 3/32 (20060101); G01N
29/46 (20060101); G01N 29/07 (20060101); G01N
29/04 (20060101); G01N 29/44 (20060101); G01N
003/00 (); G01H 013/00 () |
Field of
Search: |
;73/574-575,579,581-583,594,786-789,801,587 ;367/13 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Kwok; Helen
Attorney, Agent or Firm: Kasischke; James M. Oglo; Michael
F. Nasser; Jean-Paul A.
Government Interests
STATEMENT OF GOVERNMENT INTEREST
The invention described herein may be manufactured and used by or
for the Government of the United States of America for governmental
purposes without the payment of any royalties thereon or therefore.
Claims
What is claimed is:
1. A method of determining structural properties of a flexural beam
comprising the steps of: securing at least seven accelerometers
spaced approximately equidistant from each other along a length of
said beam; providing a transverse vibrational input to said beam at
a base; measuring seven frequency domain transfer functions of
displacement from said secured accelerometers; and estimating a
flexural wavenumber from said seven frequency domain transfer
functions; wherein said seven frequency domain transfer functions
comprise: ##EQU34##
2. The method of claim 1 further comprising the step of securing at
least one accelerometer to said base.
3. The method of claim 1 further comprising securing said beam to a
shaker table using a spring and a dashpot disposed at both a first
and a second end of said beam.
4. The method of claim 3 further comprising the step of securing at
least one accelerometer to said base.
5. The method of claim 1 wherein said base is a shaker table and
further comprising: securing a first end of said beam to the shaker
table using a spring and a dashpot; and securing a second end of
said beam to a fixed object by a pinned connection.
6. The method of claim 5, further comprising the step of securing
at least one accelerometer to said base.
7. The method of claim 1 wherein said base is a shaker table and
further comprising the steps of: securing a first end of said beam
to the shaker table using a spring and a dashpot; and securing a
second end of said beam to a fixed object using a spring and a
dashpot.
8. The method of claim 7 further comprising the step of securing at
least one accelerometer to said base.
9. The method of claim 1 wherein said base is a shaker table and
further comprising the steps of: securing a first end of maid beam
directly to the shaker table; and securing a second end of maid
beam to a fixed object by a pinned connection.
10. The method of claim 9 further comprising the step of securing
at least one accelerometer to said base.
11. The method of claim 1 further comprising the step of
determining a complex valued modulus of elasticity at each
frequency using said flexural wavenumber.
12. The method of claim 1 further comprising the step of
determining wave property coefficient using said flexural
wavenumber.
Description
BACKGROUND OF THE INVENTION
(1) Field of the Invention
This invention relates to the field of structural properties, and
in particular to the determination of the complex flexural
wavenumber, corresponding wave propagation coefficients, and
boundary condition parameters of a beam subjected to transverse
motion.
(2) Description of the Prior Art
By way of example of the state of the art, reference is made to the
following papers, which are incorporated herein by reference. Not
all of these references may be deemed to be relevant prior art.
D. M. Norris, Jr., and W. C. Young, "Complex Modulus Measurements
by Longitudinal Vibration Testing," Experimental Mechanics, Volume
10, 1970, pp. 93-96.
W. M. Madigosky and G. F. Lee, "Improved Resonance Technique for
Materials Characterization," Journal of the Acoustical Society of
America, Volume 73, Number 4, 1983, pp. 1374-1377.
S. L. Garrett, "Resonant Acoustic Determination of Elastic Moduli,"
Journal of the Acoustical Society of America, Volume 88, Number 1,
1990, pp. 210-220.
I. Jimeno-Fernandez, H. Uberall, W. M. Madigosky, and R. B.
Fiorito, "Resonance Decomposition for the Vibratory Response of a
Viscoelastic Rod," Journal of the Acoustical Society of America,
Volume 91, Number 4, Part 1, April 1992, pp. 2030-2033.
G. F. Lee and B. Hartmann, "Material Characterizing System," U.S.
Pat. No. 5,363,701, Nov. 15, 1994.
G. W. Rhodes, A. Migliori, and R. D. Dixon, "Method for Resonant
Measurement," U.S. Pat. No. 5,495,763, Mar. 5, 1996.
R. F. Gibson and E. O. Ayorinde, "Method and Apparatus for
Non-Destructive Measurement of Elastic Properties of Structural
Materials," U.S. Pat. No. 5,533,399, Jul. 9, 1996.
B. J. Dobson, "A Straight-Line Technique for Extracting Modal
Properties From Frequency Response Data," Mechanical Systems and
Signal Processing, Volume 1, 1987, pp. 29-40.
C. Minas and D. J. Inman, "Matching Finite Element Models to Modal
Data," Journal of Vibration and Acoustics, Volume 112, Number 1,
1990, pp. 84-92,
T. Pritz, "Transfer Function Method for Investigating the Complex
Modulus of Acoustic Materials: Rod-Like Specimen," Journal of Sound
and Vibration, Volume 81, 1982, pp. 359-376.
W. M. Madigosky and G. F. Lee, "Instrument for Measuring Dynamic
Viscoelastic Properties," U.S. Pat. No. 4,352,292, Oct. 5,
1982.
W. M. Madigosky and G. F. Lee, "Method for Measuring Material
Characteristics," U.S. Pat. No. 4,418,573, Dec. 6, 1983.
W. Madigosky, "In Situ Dynamic Material Property Measurement
System," U.S. Pat. No. 5,365,457, Nov. 15, 1994.
J. G. McDaniel, P. Dupont, and L. Salvino, "A Wave Approach to
Estimating Frequency-Dependent Damping Under Transient Loading"
Journal of Sound and Vibration, Volume 231(2), 2000, pp.
433-449.
J. Linjama and T. Lahti, "Measurement of Bending wave reflection
and Impedance in a Beam by the Structural Intensity Technique"
Journal of Sound and Vibration, Volume 161(2), 1993, pp.
317-331.
L. Koss and D. Karczub, "Euler Beam Bending Wave Solution
Predictions of dynamic Strain Using Frequency Response Functions "
Journal of Sound and Vibration, Volume 184(2), 1995, pp.
229-244.
Measuring the flexural properties of beams is important because
these parameters significantly contribute to the static and dynamic
response of structures. In the past, resonant techniques have been
used to identify and measure longitudinal properties. These methods
are based on comparing the measured eigenvalues of a structure to
predicted eigenvalues from a model of the same structure. The model
of the structure must have well-defined (typically closed form)
eigenvalues for this method to work. Additionally, resonant
techniques only allow measurements at natural frequencies.
Comparison of analytical models to measured frequency response
functions is another method used to estimate stiffness and loss
parameters of a structure. When the analytical model agrees with
one or more frequency response functions, the parameters used to
calculate the analytical model are considered accurate. If the
analytical model is formulated using a numerical method, a
comparison of the model to the data can be difficult due to the
dispersion properties of the materials.
Another method to measure stiffness and loss is to deform the
material and measure the resistance to the indentation. This method
can physically damage the specimen if the deformation causes the
sample to enter the plastic region of deformation.
SUMMARY OF THE INVENTION
Accordingly, one objective of the present invention is to measure
flexural wavenumbers.
Another objective of the present invention is to measure flexural
wave propagation coefficients.
A further objective of the present invention is to measure Young's
modulus when the beam is undergoing transverse motion.
Yet another objective of the present invention is to measure the
boundary stiffness and dampening values when the beam is vibrated
transversely.
The foregoing objects are attained by the method and system of the
present invention. The present invention features a method of
determining structural properties of a flexural beam mounted to a
base. The method comprises securing a plurality of accelerometers
spaced approximately equidistant from each other along a length of
a beam. One accelerometer can be secured to the base. An input is
provided to the beam. Seven frequency domain transfer functions of
displacement are measured from the accelerometers secured to the
beam. The flexural wavenumber is estimated from the seven frequency
domain transfer functions.
The seven frequency domain transfer functions of displacement
include the following equations: ##EQU1##
The flexural wavenumber is determined using the following
equations: ##EQU2##
and said imaginary part comprises: ##EQU3##
Using the flexural wavenumber and various equations disclosed
within the present invention, the complex valued modulus of
elasticity can be determined at each frequency, as well as the wave
property coefficient, and the boundary parameters.
Thus, this invention has the advantages that all measurements can
be calculated at every frequency that a transfer function
measurement is made. They do not depend on system resonance's or
curve fitting to transfer functions. The calculation from transfer
function measurement to calculation of all system parameters is
exact, i.e., no errors are introduced during this process.
Furthermore, the measurements can be calculated without adverse
consequences to the tested beam.
BRIEF DESCRIPTION OF THE DRAWINGS
These and other features and advantages of the present invention
will be better understood in view of the following description of
the invention taken together with the drawings wherein:
FIG. 1 is a schematic block diagram of a conventional testing
system including two springs and two dashpots attached to a shaker
table;
FIG. 2 is a schematic block diagram of a conventional testing
system including one spring and one dashpot attached to a shaker
table;
FIG. 3 is a schematic block diagram of a conventional testing
system including two springs and two dashpots, one of which is
attached to a shaker table;
FIG. 4 is a schematic block diagram of a conventional testing
system wherein the beam is attached directly to a shaker table;
FIG. 5A is a graph of the magnitude of a typical transfer function
of a beam;
FIG. 5B is a graph of the phase angle of a typical transfer
function of a beam;
FIG. 6 is a graph of the function s versus frequency;
FIG. 7A is a graph of the real part of a flexural wavenumber versus
frequency;
FIG. 7B is a graph of the imaginary part of a flexural wavenumber
versus frequency;
FIGS. 8-11 are graphs of the wave propagation coefficients versus
frequency;
FIG. 12 is a graph of the real and imaginary parts of the Young's
Modulus versus frequency;
FIG. 13 is a graph of the boundary conditions of the system shown
in FIG. 1 versus frequency; and
FIG. 14 is a graph of the boundary conditions of the system shown
in FIG. 2 versus frequency.
DESCRIPTION OF THE PREFERRED EMBODIMENT
The method and system, according to the present invention, is used
to develop and measure complex flexural wavenumbers and the
corresponding wave propagation coefficients of a beam undergoing
transverse motion. An inverse method has been developed using seven
transfer function measurements. These seven transfer function
measurements are combined to yield closed form values of flexural
wavenumber and wave propagation coefficients at any given test
frequency. Finally, Young's modulus, spring stiffnesses, dashpot
damping values, and boundary condition parameters, among other
parameters, are calculated from the flexural wavenumber and wave
propagation coefficients.
According to an exemplary test configuration 10, FIG. 1, a shaker
table 12 initiates transverse motion 14 into a beam 16. The beam 16
is connected to the shaker table 12 with a spring 18 and dashpot 20
at each end 22. FIG. 1 represents a double translational spring and
damper input configuration. Other test configurations are also
possible, including the shaker table 12 inputting energy into only
one end 22 of the beam 16 with the other end terminated to ground
24 directly, as shown in FIG. 2, or terminated to ground 24 with a
spring 18 and dashpot 20, as shown in FIG. 3, or terminated to
ground 24 and the shaker 12 directly, as shown in FIG. 4. FIG. 2
represents a single translational spring and damper input
configuration with the other end pinned. FIG. 3 represents a single
translational spring and damper input configuration with the other
end having a translational spring and damper. FIG. 4 represents a
single pin input configuration with the other end pinned. These
approaches are intended for use when a beam 16 is to undergo motion
in the transverse direction 14. This system typically arises in
cars, ships, aircraft, bridges, buildings and other common
structures.
In any of the embodiments shown in FIGS. 1-4 sensors 26 such as
accelerators are positioned equally along beam 22. As discussed
above, a minimum of seven such sensors 22 are required. Optionally,
a reference sensor 28 can be joined to shaker table 12 to read the
input motion 14. The input motion 14 can also be read directly from
the shaker table 12 controls.
For simplicity, the present invention will be described as it
relates to the derivation of the linear equations of motion of the
system with a spring 18 and dashpot 20 boundary condition at each
end 22, but this is for exemplary purposes only, and is not
intended to be a limitation.
The system model of the beam is the Bernoulli-Euler beam equation
written as ##EQU4##
where x is the distance along the length of the beam in meters, t
is time in seconds, u is the displacement of the beam in the
(transverse) y direction in meters, E is the unknown
frequency-dependent, complex Young's modulus (N/m.sup.2), I is the
moment of inertia (m.sup.4), .rho. is the density (kg/m.sup.3), and
A.sub.b is the cross-sectional area of the beam (m.sup.2). Implicit
in equation (1) is the assumption that plane sections remain planar
during bending (or transverse motion). Additionally, Young's
modulus, the moment of inertia, the density, and the cross
sectional area are constant across the entire length of the beam.
The displacement is modeled as a steady state response and is
expressed as
where .omega. is the frequency of excitation (rad/s), U(x,.omega.)
is the temporal Fourier transform of the transverse displacement,
and i is the square root of -1. The temporal solution to equation
(1), derived using equation (2) and written in terms of
trigonometric functions, is
where A(.omega.), B(.omega.), C(.omega.), and D(.omega.) are wave
propagation coefficients and .alpha.(.omega.) is the flexural
wavenumber given by ##EQU5##
For brevity, the .omega. dependence is omitted from the wave
propagation coefficients and the flexural wavenumber during the
remainder of the disclosure and .alpha.(.omega.) is references as
.alpha.. Note that equations (3) and (4) are independent of
boundary conditions, and the inverse model developed in the next
section does not need boundary condition specifications. Boundary
conditions are chosen, however, to show that the boundary
parameters can be estimated and to run a realistic simulation.
One of the most typical test configurations is the beam mounted to
shock mounts on each end that are attached to a shaker table that
generates a vibrational input, as shown in FIG. 1. Using the middle
of the beam as the coordinate system origin, these boundary
conditions are modeled as ##EQU6##
where
which is the input into the system from the shaker table.
Inserting equation (3) into equation (5), (6), (7), (8), and (9)
yields the solution to the wave propagation coefficients. Inserting
these back into equation (3) is the displacement of the system, and
is sometimes called the forward solution. The wave coefficient A is
##EQU7##
where ##EQU8##
The wave coefficient B is ##EQU9##
where ##EQU10##
The wave coefficient C is ##EQU11##
where ##EQU12##
The wave coefficient D is ##EQU13##
where ##EQU14##
These coefficients are used for the simulation below. If the beam
model corresponds to FIGS. 2, 3, or 4, then the boundary conditions
given in equations (5)-(8) change slightly as do the wave
propagation coefficients.
Equation (3) has five unknowns and is nonlinear with respect to the
unknown flexural wavenumber. It will be shown that using seven
independent, equally spaced measurements, that the five unknowns
can be estimated with closed form solutions. Furthermore, in the
next section, it will be shown that the components that comprise
the beams mounting system can also be estimated. Seven frequency
domain transfer functions of displacement are now measured. These
consist of the measurement at some location divided by a common
measurement. Typically this would be an accelerometer at a
measurement location and an accelerometer at the base of a shaker
table. These seven measurements are set equal the theoretical
expression given in equation (3) and are listed as ##EQU15##
where .delta. is the sensor to sensor separation distance (m) and
V.sub.0 (.omega.) is the reference measurement. Note that the units
of the transfer functions given in equations (20)-(26) are
dimensionless.
Equation (22) is now subtracted from equation (24), equation (21)
is subtracted from equation (25), and equation (20) is subtracted
from equation (26), yielding the following three equations:
##EQU16##
Equations (27), (28), and (29) are now combined to give
##EQU17##
Equation (22) is now added to equation (24) and equation (21) is
added to equation (25), yielding the following two equations:
##EQU18##
and ##EQU19##
Equations (23), (31), and (32) are now combined to yield the
following equation: ##EQU20##
Equation (30) and (33) are now combined, and the result is a
binomial expression with respect to the cosine function, and is
written as
where
and
Equation (34) is now solved using ##EQU21##
where .phi. is typically a complex number. Equation (38) is two
solutions to equation (34). One, however, will have an absolute
value less than one and that is the root that is further
manipulated. The inversion of equation (38) allows the complex
flexural wavenumber .alpha. to be solved as a function of .phi. at
every frequency in which a measurement is made. The solution to the
real part of .alpha. is ##EQU22##
where ##EQU23##
n is a non-negative integer and the capital A denotes the principal
value of the inverse cosine function. The value of n is determined
from the function s, which is a periodically varying cosine
function with respect to frequency. At zero frequency, n is 0.
Every time s cycles through .pi. radians (180 degrees), n is
increased by 1. When the solution to the real part of .alpha. is
found, the solution to the imaginary part of .alpha. is then
written as ##EQU24##
Once the real and imaginary parts of wavenumber .alpha. are known,
the complex valued modulus of elasticity can be determined at each
frequency with ##EQU25##
assuming that the density, area, and moment of inertia of the beam
are known. Equations (20)-(42) produce an estimate Young's modulus
at every frequency in which a measurement is conducted.
Additionally, combining equations (27) and (28) yields
##EQU26##
and ##EQU27##
Combining equations (23) and (31) yields ##EQU28##
and ##EQU29##
Equations (43)-(46) are the estimates of the complex wave
propagation coefficients. These are normally considered less
important than the estimate of the flexural wavenumber. It will be
shown, however, that these coefficients can be used to estimate the
boundary condition parameters of the beam.
Inserting equations (2), (3), (4), and (9) into equation (6) and
solving for the boundary parameters at x=-L/2 yields ##EQU30##
and ##EQU31##
Similarly, inserting equations (2), (3), (4), and (9) into equation
(8) and solving for the boundary parameters at x=L/2 yields
##EQU32##
and ##EQU33##
Thus, once the flexural wavenumber and wave coefficients are
estimated, the properties of the springs and dashpots at the
boundaries can be calculated.
Numerical simulations conducted to determine the effectiveness of
this method use the following parameters to define a baseline
problem: Re(E)=(3.multidot.10.sup.10 +10.sup.7 f) N/m.sup.2,
Im(E)=(3.multidot.10.sup.9 +10.sup.6 f) N/m.sup.2, .rho.=5000
kg/m.sup.3, A.sub.b =0.02 m.sup.2, I=6.67.times.10.sup.-5 m.sup.4,
L=3 m, .delta.=0.5 m, k.sub.1 =50000 N/m, c.sub.1 =4000
N.multidot.s/m, k.sub.2 =60000 N/m, and c.sub.2 =5000
N.multidot.s/m where f is frequency in Hz. FIGS. 5A and 5B
represent a typical transfer function of the beam displacement
measured at x=0 m, which is the middle of the beam, divided by base
displacement. The top plot, FIG. 5A, is the magnitude versus
frequency and the bottom plot, FIG. 5B, is the phase angle versus
frequency. This figure was constructed by inserting the above
parameters into equations (3), (4), (10), (11), (12), (13), (14),
(15), (16), (17), (18), and (19) and calculating the solution (a
forward model).
FIG. 6 graphs the function s versus frequency. It was calculated by
inserting the left-hand side of equations (20)-(26) into equations
(34)-(40) and represents the first step of the inverse method
calculations. FIGS. 7A and 7B represent the flexural wavenumber
versus frequency. The top plot, FIG. 7A, is the real part and the
bottom plot, FIG. 7B, is the imaginary part. The values created
using equation (4) (the forward solution) are shown as solid lines
and the values calculated (or estimated) using equations (34)-(41)
(the inverse solution) are shown with x's and o's. Note that there
is total agreement among the forward and inverse solutions. FIGS.
8-11 are the wave propagation coefficients A, B, C, and D versus
frequency, respectively. The top plots are the magnitudes and the
bottom plots are the phase angles. The values created using
equation (10)-(19) (the forward solution) are shown as solid lines
and the values calculated using equations (43)-(46) (the inverse
solution) are shown with x's and o's. FIG. 12A and FIG. 12B graph
the real and imaginary parts of Young's modulus versus frequency.
The actual values are shown as solid lines and the values
calculated using equation (42) are shown with x's and o's. FIG. 13
is the boundary condition parameters of mount one versus frequency.
The top plot is the stiffness and the bottom plot is the damping.
The actual values are shown as solid lines and the values
calculated using equations (47) and (48) are shown with x's and
o's. FIG. 14 is the boundary condition parameters of mount two
versus frequency. The top plot, FIG. 14A, is the stiffness and the
bottom plot, FIG. 14B, is the damping. The actual values are shown
as solid lines and the values calculated using equations (49) and
(50) are shown with x's and o's.
In light of the above, it is therefore understood that within the
scope of the appended claims, the invention may be practiced
otherwise than as specifically described.
* * * * *